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On the Lebesgue constant of the Floater-Hormann rational interpolants Stefano De Marchi Department of Mathematics University of Padova Fribourg, November 3, 2015 Joint work with L. Bos (Verona), K. Hormann (Lugano), G. Klein (Fribourg),

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On the Lebesgue constant of the Floater-Hormann rational interpolants ∗

Stefano De Marchi

Department of Mathematics University of Padova

Fribourg, November 3, 2015

∗Joint work with L. Bos (Verona), K. Hormann (Lugano), G. Klein (Fribourg), J. Sidon (TelAviv) Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 1 / 47

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Outline

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 2 / 47

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Motivations

Known things and aim Known Michael S. Floater and Kai Hormann, Barycentric rational interpolation with no poles and high rates of approximation, Numer. Math. 107(2) (2007), 315–331. Floater and Hormann rational interpolants, FHRI, is a family of rational interpolants that perform rational interpolations on equispaced and non-equispaced points . From their paper... “it seems to be perfectly stable in practice”... but nothing was proved about its stability. The Lebesgue constant measures the stability of an interpolation process. FHRI is also on Numerical Recepies, section 3.4.1

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 3 / 47

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Motivations

Known things and aim Known Michael S. Floater and Kai Hormann, Barycentric rational interpolation with no poles and high rates of approximation, Numer. Math. 107(2) (2007), 315–331. Floater and Hormann rational interpolants, FHRI, is a family of rational interpolants that perform rational interpolations on equispaced and non-equispaced points . From their paper... “it seems to be perfectly stable in practice”... but nothing was proved about its stability. The Lebesgue constant measures the stability of an interpolation process. FHRI is also on Numerical Recepies, section 3.4.1 Aim What’s about the growth of the Lebesgue constants for the FHRI?

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 3 / 47

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The interpolant

Lagrange form of the interpolant

Given a function f : [a, b] → R, let g be its interpolant at the n + 1 (equispaced) interpolation points a = x0 < x1 < · · · < xn = b. Given a set of basis functions bi which satisfy the Lagrange property bi(xj) = δij =

  • 1,

if i = j, 0, if i = j, the interpolant g can be written as g(x) =

n

  • i=0

bi(x)f (xi) =

n

  • i=0

bi(x)yi .

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 4 / 47

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The interpolant

Barycentric form of the interpolant

Interpolation of 2 data points x0, x1, g(x) = 1

i=0 λi(x)yi

1

i=0 λi(x)

, λi(x) = (−1)i x − xi i = 0, 1 and bi(x) = λi(x) 1

i=0 λi(x)

.

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 5 / 47

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The interpolant

Barycentric form of the interpolant

Interpolation of 2 data points x0, x1, g(x) = 1

i=0 λi(x)yi

1

i=0 λi(x)

, λi(x) = (−1)i x − xi i = 0, 1 and bi(x) = λi(x) 1

i=0 λi(x)

. Interpolation of n + 1 data points g(x) = n

i=0 λi(x)yi

n

i=0 λi(x) , λi(x) =

(−1)i (x − xi) .

n

  • i=0

λi(x) = 1 x − x0

>0

+ −1 x − x1 + 1 x − x2

  • >0

+ −1 x − x3 + · · ·

  • >0

x0 < x < x1

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 5 / 47

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The interpolant

The Floater-Hormann Rational Interpolant (FHRI) The construction of FHRI, is very simple. Let 0 ≤ d ≤ n. For each i = 0, 1, . . . , n − d let pi denote the unique polynomial of degree at most d that interpolates a function f at d + 1 pts xi, . . . , xi+d Then g(x) =

n−d

  • i=0

λi(x)pi(x)

n−d

  • i=0

λi(x) (1) where λi(x) = (−1)i (x − xi) · · · (x − xi+d).

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 6 / 47

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The interpolant

The Floater-Hormann Rational Interpolant (FHRI) The construction of FHRI, is very simple. Let 0 ≤ d ≤ n. For each i = 0, 1, . . . , n − d let pi denote the unique polynomial of degree at most d that interpolates a function f at d + 1 pts xi, . . . , xi+d Then g(x) =

n−d

  • i=0

λi(x)pi(x)

n−d

  • i=0

λi(x) (1) where λi(x) = (−1)i (x − xi) · · · (x − xi+d).

Thus, g is a local blending of the polynomial interpolants p0, . . . , pn−d with λ0, . . . , λn−d acting as the blending functions. Notice: for d = n we get the classical polynomial interpolation. Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 6 / 47

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The interpolant

Basis functions

Assume [a, b] = [0, 1] and equispaced interpolation pts xi = i/n, i = 0, . . . , n.

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 7 / 47

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The interpolant

Basis functions

Assume [a, b] = [0, 1] and equispaced interpolation pts xi = i/n, i = 0, . . . , n. As basis functions we take bi(x) = (−1)iβi x − xi

  • n
  • j=0

(−1)jβj x − xj , i = 0, . . . , n (2) with β0, . . . , βn positive weights defined as βj =      j

k=0

d

k

  • ,

if j ≤ d, 2d, if d ≤ j ≤ n − d, βn−j, if j ≥ n − d. (3)

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 7 / 47

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The interpolant

The weights βs

d = 0† 1, 1, . . . , 1, 1 d = 1‡ 1, 2, 2 . . . , 2, 2, 1 d = 2 1, 3, 4, 4, . . . , 4, 4, 3, 1 d = 3 1, 4, 7, 8, 8, . . . , 8, 8, 7, 4, 1 d = 4 1, 5, 11, 15, 16, 16, . . . , 16, 16, 15, 11, 5, 1

†Berrut’s rational interpolant ‡d ≥ 1 Floater-Hormann’s rational interpolant Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 8 / 47

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The interpolant

Some plots of the basis functions

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 9 / 47

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The interpolant

Basis functions

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 10 / 47

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The interpolant

Interpolation Figure: FHRI compared with a cubic spline on 11 equispaced points for the function |x|, x ∈ [−1, 1]

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 11 / 47

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The interpolant

Properties of the FHRI (cf. [FH, NumMath2007])

  • 1. The FHRI can be written in barycentric form.

Indeed, in (1), letting wi = (−1)iβi, for the numerator we have

n−d

  • i=0

λi(x)pi(x) =

n

  • k=0

wk x − xk f (xk) where wk =

  • i∈Ik

(−1)i

i+d

  • j=k,j=i

1 xk − xj Ik = {i ∈ J, k − d ≤ i ≤ k}, J := {0, ..., n − d}. Similarly for the denominator

n−d

  • i=0

λi(x) =

n

  • k=0

wk x − xk It is a rational function of degree (n,n-d)

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 12 / 47

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The interpolant

Properties of the FHRI (continue)

  • 2. The rational interpolant g(x) has no real poles. For d = 0 was proved by

Berrut in 1998.

  • 3. The interpolant reproduces polynomials of degree at most d, while does not

reproduce rational functions (like Runge function)

  • 4. Approximation error order O(hd+1) (for f ∈ Cd+2[0, 1]), also for

non-equispaced points.

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 13 / 47

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The Lebesgue Constant The case d = 0

Lebesgue constant when d = 0

Remember: when d = 0, βj = 1, ∀j. We will derive upper and lower bounds for the Lebesgue function Λn(x) =

n

  • i=0

|bi(x)| =

n

  • i=0

βi |x − xi|

  • n
  • j=0

(−1)jβj x − xj

  • .

(4) so that we can estimate Λ = max

x∈[0,1] Λn(x)

(Lebesgue constant) .

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 14 / 47

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The Lebesgue Constant The case d = 0

Lebesgue constant when d = 0

Remember: when d = 0, βj = 1, ∀j. We will derive upper and lower bounds for the Lebesgue function Λn(x) =

n

  • i=0

|bi(x)| =

n

  • i=0

βi |x − xi|

  • n
  • j=0

(−1)jβj x − xj

  • .

(4) so that we can estimate Λ = max

x∈[0,1] Λn(x)

(Lebesgue constant) . Theorem (BDeMH, JCAM11) For any n ≥ 1, we have cn log(n + 1) ≤ Λ ≤ 2 + log(n). where cn = 2n/(4 + nπ) (limn→∞ cn = 2/π).

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 14 / 47

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The Lebesgue Constant The case d = 0

Case d = 0: lower bound We assume that the interpolation interval is [0, 1], so that the nodes are equally spaced xj = jh = j · 1/n, j = 0, . . . , n.

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 15 / 47

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The Lebesgue Constant The case d = 0

Case d = 0: lower bound We assume that the interpolation interval is [0, 1], so that the nodes are equally spaced xj = jh = j · 1/n, j = 0, . . . , n. Our goal is bounding below Λn(x) =

n

  • j=0

1 |x − j/n|

  • n
  • j=0

(−1)j x − j/n

  • =

n

  • j=0

1 |2nx − 2j|

  • n
  • j=0

(−1)j 2nx − 2j

  • := N(x)

D(x). (5) by bounding N(x) from below and D(x) from above

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 15 / 47

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The Lebesgue Constant The case d = 0

The Lebesgue function for d = 0 on equispaced points Figure: Lebesgue functions on [0,1]: n=10 (11 pts) (left) and n=11 (right). The black and red signs indicate the points where the max is taken

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 16 / 47

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The Lebesgue Constant The case d = 0

Case d = 0: lower and upper bounds for N(x) and D(x) Assume n = 2k and let x∗ = (n + 1)/2n = 1/2 + 1/(2n). Bounds [JCAM2011] N(x∗) ≥ 1 2 (ln(2k + 3) + ln(2k + 1)) ≥ ln(2k + 1) = ln(n + 1) D(x∗) ≤ π 2 + 2 2k + 1 = π 2 + 2 n + 1 .

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 17 / 47

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The Lebesgue Constant The case d = 0

Case d = 0: lower and upper bounds for N(x) and D(x) Assume n = 2k and let x∗ = (n + 1)/2n = 1/2 + 1/(2n). Bounds [JCAM2011] N(x∗) ≥ 1 2 (ln(2k + 3) + ln(2k + 1)) ≥ ln(2k + 1) = ln(n + 1) D(x∗) ≤ π 2 + 2 2k + 1 = π 2 + 2 n + 1 . Hence, Λn(x∗) ≥ 2 ln(n + 1) π +

4 n+1

.

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 17 / 47

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The Lebesgue Constant The case d = 0

Case d = 0. Lower bound for Λ The same is true when n is odd considering x∗ = 1/2, instead.

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 18 / 47

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The Lebesgue Constant The case d = 0

Case d = 0. Lower bound for Λ The same is true when n is odd considering x∗ = 1/2, instead. In summary, for any n ∈ N Λ = max

0≤x≤1 Λn(x) ≥ 2 ln(n + 1)

π +

4 n+1

≥ 2 ln(n + 1) π + 4

n

= cn ln(n + 1) . where cn =

2 n 4+πn .

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 18 / 47

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The Lebesgue Constant The case d = 0

Case d = 0: upper and lower bounds for N(x) and D(x) Let xk < x < xk+1 for some k and let Nk(x) and Dk(x), N, D on the interval. Bounds on the k-th interval [JCAM2011] Nk(x) ≤ 1 n + 1 2n ln(n) Dk(x) ≥ 1 2n . These bounds hold regardless the parity of n and k.

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 19 / 47

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The Lebesgue Constant The case d = 0

Case d = 0: upper and lower bounds for N(x) and D(x) Let xk < x < xk+1 for some k and let Nk(x) and Dk(x), N, D on the interval. Bounds on the k-th interval [JCAM2011] Nk(x) ≤ 1 n + 1 2n ln(n) Dk(x) ≥ 1 2n . These bounds hold regardless the parity of n and k. Combining the bounds for numerator and denominator yields Λ = max

k=0,...,n

  • max

xk<x<xk+1 Λk(x)

  • ≤

1 n + 1 2n log(n) 1 2n

= 2 + log(n).

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 19 / 47

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The Lebesgue Constant The case d = 0

The Lebesgue constant for d = 0 on uniform pts Figure: Lebesgue constant compared with its lower and upper bounds.

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 20 / 47

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The Lebesgue Constant The case d ≥ 1

Lebesgue constant: case d ≥ 1 Notice that βj ≤ 2d, ∀ j

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 21 / 47

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The Lebesgue Constant The case d ≥ 1

Lebesgue constant: case d ≥ 1 Notice that βj ≤ 2d, ∀ j For xk < x < xk+1 the numerator can be bound as follows Nk(x) = (x − xk)(xk+1 − x)

n

  • j=0

βj |x − xj| ≤ 2d(x − xk)(xk+1 − x)

n

  • j=0

1 |x − xj| ≤ 2d 1 n + 1 2n log(n)

  • ,

(6) = ⇒ that holds for any k.

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 21 / 47

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The Lebesgue Constant The case d ≥ 1

The denominator Fundamental observation (−1)jβj = wj d! hd (7) Then, Dk(x) = (x − xk)(xk+1 − x)

  • n
  • j=0

wj x − xj

  • d!hd .

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 22 / 47

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The Lebesgue Constant The case d ≥ 1

The denominator Fundamental observation (−1)jβj = wj d! hd (7) Then, Dk(x) = (x − xk)(xk+1 − x)

  • n
  • j=0

wj x − xj

  • d!hd .

Since [FH, NumerMath2007],

n

  • j=0

wj x − xj =

n−d

  • i=0

λi(x) = ⇒

  • n
  • j=0

wj x − xj

  • ≥ |λk(x)| .

Then, Dk(x) = (x−xk)(xk+1−x)|λk(x)|d!hd = d!hd k+d

l=k+2(xl − x)

≥ d!hd k+d

l=k+2(xl − xk)

= 1 n .

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 22 / 47

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The Lebesgue Constant The case d ≥ 1

The lower bound

Theorem (Klein, Dec. 2010) Let d ≥ 2 then, Λ ≥ (2d + 1)!! 4(d + 1)! log n d − 1

  • .

Theorem (Bos, Dec. 2010) Let d ≥ 1 then, Λ ≥ 2 π log(n + 2 − 2d). Note: this latter is better for d = 1.

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 23 / 47

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The Lebesgue Constant The case d ≥ 1

The Lebesgue constant bounds for d ≥ 1

Theorem Let d > 1 Then, (2d + 1)!! 4(d + 1)! log n d − 1

  • ≤ Λ ≤ 2d−1

2 + log(n)

  • .

while for d = 1 2 π log(n) ≤ Λ ≤ 2 + log(n).

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 24 / 47

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The Lebesgue Constant Numerical results

Lebesgue functions

Figure: Lebesgue function for d = 1 (left) and d = 3 (right).

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 25 / 47

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The Lebesgue Constant Numerical results

Lebesgue constants growth: I Figure: Lebesgue constant growth d = 1 (left) and d = 3 (right).

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 26 / 47

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The Lebesgue Constant Numerical results

Lebesgue constant growth: II Figure: Lebesgue constant growth d = 8 (left) and d = 16 (right).

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 27 / 47

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Quasi-equispaced case

Quasi-equidistant nodes Equispaced nodes perturbed by a randomly chosen δ ∈ (0, 1/2), that is xj = j + δj, j = 0, . . . , n . We also assume that there exist M ≥ 1 (independent on n) s.t. set h := max0≤j≤n−1(xj+1 − xj) and h∗ := min0≤j≤n−1(xj+1 − xj) then h h∗ ≤ M .

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 28 / 47

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Quasi-equispaced case

Quasi-equidistant nodes

Lemma (Bounds on the weights, HKDeM2012) Wk ≤ |wk| ≤ MdWk where Wk = 1 hdd!

  • i∈Jk

d k − i

  • , k = 0, 1, . . . , n

Moreover Wk ≤ 2d hdd! := W with equality iff d ≤ k ≤ n − d.

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 29 / 47

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Quasi-equispaced case

Quasi-equidistant nodes Theorem (Bounds on the Λn, HKDeM2012) Λn ≥ 1 2d+2Md+1 2d + 1 d

  • ·

   (2 + log(2n + 1)) d=0 log( n

d − 1)

d ≥ 1 Λn ≤ (2 + M log(n)) ·   

3 4M,

d=0 2d−1Md, d ≥ 1

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 30 / 47

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Quasi-equispaced case

Lebesgue functions for quasi-equidistant points Figure: Lebesgue functions for 30 quasi-equidistant points perturbed at first, central, 5th and central points respectively, for d = 2

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 31 / 47

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Well-spaced nodes

Definition The set X = (Xn)n≥0 is a family of well-spaced nodes, if there exist R, C ≥ 1 (independent on n) so that xk+1 − xk xk+1 − xj ≤ C k + 1 − j , j = 0, ..., k k = 0, ..., n − 1, xk+1 − xk xj − xk ≤ C j − k , j = k + 1, ..., n k = 0, ..., n − 1, 1 R ≤ xk+1 − xk xk − xk−1 ≤ R, k = 1, ..., n − 1, hold for each set Xn.

  • Note. When the nodes are equispaced R = C = 1. The definition include also

Chebyshev and extended Chebyshev nodes.

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 32 / 47

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Well-spaced nodes

Lebesgue constant growth for d = 0 Theorem (Bounds on the Λ and d = 0, BDeMHS2013) If X = (Xn)n≥0 is a family of well-spaced nodes then Λ(Xn) ≤ (R + 1)(1 + 2C log(n)) = c log(n), n ≥ 2 .

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 33 / 47

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Well-spaced nodes

How to get well-spaced nodes? Definition We say that a function F ∈ C[0, 1] is a distribution function if it is a strictly increasing bijection on the interval [0, 1]

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 34 / 47

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Well-spaced nodes

How to get well-spaced nodes? Definition We say that a function F ∈ C[0, 1] is a distribution function if it is a strictly increasing bijection on the interval [0, 1] Given F we may associate the node sets xk := F(k/n), k = 0, . . . , n .

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 34 / 47

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Well-spaced nodes

How to get well-spaced nodes? Definition We say that a function F ∈ C[0, 1] is a distribution function if it is a strictly increasing bijection on the interval [0, 1] Given F we may associate the node sets xk := F(k/n), k = 0, . . . , n . = ⇒ To generate points that realize the bound of the previous Theorem, F has to be as follows Definition We say that a distribution is regular, if F ∈ C′[0, 1] and F ′ has a finite number of zeros T = {t1, . . . , tl} ⊂ [0, 1] with finite multiplicities.

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 34 / 47

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Well-spaced nodes

Properties of regular distributions Proposition Let F be a regular distribution function. Then there exists a constant C > 0 such that F[x, y] F[x, z] ≤ C for all x, y, z ∈ [0, 1] s.t. x > y ≥ z. Proposition Let F be a regular distribution function. Then there exist an ǫ > 0 and R > 0 such that 1 R ≤ F[x, x + s] F[x − s, x] ≤ R for all s ∈ [0, ǫ] and x ∈ [s, 1 − s].

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 35 / 47

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Well-spaced nodes

Examples of regular distributions

1

F1(x) = x, that generates equispaced points.

2

F2(x) = log(1 + x(e − 1)), that generates logarithmically distributed points. Note that F2 is regular since F ′

2 > 0.

3

F3(x) = (1 − cos(πx))/2, that is regular since F ′

3 has 2 simple zeros at

x1 = 0, x2 = 1. This generates the Chebyshev extrema (or Chebyshev-Lobatto points) mapped in [0, 1]. In this case, for δ = 1/2, ǫ = 1/4, we get C = 2π and R = 9π/2 and Λ(Xn) ≤ (9π/2 + 1)(1 + 4π log(n))) .

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 36 / 47

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SLIDE 50

Well-spaced nodes

Non-regular distributions

1

F4(x) =

  • x = 0

exp(1 − 1/x) 0 < x ≤ 1 which is non-regular since C∞ at x = 0

2

F5(x) = 1 2    1 − exp(1 + 1/(2x − 1)) 0 ≤ x < 1/2 1 x = 1/2 1 + exp(1 − 1/(2x − 1)) 1/2 ≤ x ≤ 1 F ′(x) = 0 in x = 1/2 with infinite multiplicity. The last is non regular for odd n while for even n seems to growth logaritmically

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 37 / 47

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SLIDE 51

Well-spaced nodes

Other non-regular nodes

We can also verify directly if a given family of nodes is well-spaced even if it is not generated by a distribution. Extended Chebyshev nodes (on [0,1]) xk = 1 2  1 − cos

  • 2k+1

2n+2π

  • cos
  • π

2n+2

 , k = 0, . . . , n

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 38 / 47

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SLIDE 52

Well-spaced nodes

Other non-regular nodes

We can also verify directly if a given family of nodes is well-spaced even if it is not generated by a distribution. Extended Chebyshev nodes (on [0,1]) xk = 1 2  1 − cos

  • 2k+1

2n+2π

  • cos
  • π

2n+2

 , k = 0, . . . , n Proposition For extended Chebyshev nodes we have xk+1 − xk xk+1 − xj ≤ π2/2 k + 1 − j , j = 0, . . . , k, k = 0, . . . , n − 1, ∀n . 1 2 ≤ xk+1 − xk xk − xk−1 ≤ 2, k = 0, . . . , n − 1, ∀n . = ⇒ they satisfy the Definition of well-spaceness with C = π2/2 and R = 2

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 38 / 47

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SLIDE 53

Well-spaced nodes

Lebesgue constant growth for EC nodes

Figure: Lebesgue constant for Berrut’s rational interpolant at n + 1 extended Chebyshev nodes for n = 1, . . . , 50. giving the bound Λ(Xn) ≤ 3 + 3π2 log(n) .

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 39 / 47

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SLIDE 54

An application

Numerical quadrature On I = [−1, 1]

1

we computed integrals with the quadrature based on the FHRI, on equispaced points at different values of n and d

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 40 / 47

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SLIDE 55

An application

Numerical quadrature On I = [−1, 1]

1

we computed integrals with the quadrature based on the FHRI, on equispaced points at different values of n and d

2

to speed up the quadrature, the quadrature weights were computed by a Gaussian quadrature rule (Gautschi software in Matlab)

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 40 / 47

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SLIDE 56

An application

Numerical quadrature On I = [−1, 1]

1

we computed integrals with the quadrature based on the FHRI, on equispaced points at different values of n and d

2

to speed up the quadrature, the quadrature weights were computed by a Gaussian quadrature rule (Gautschi software in Matlab)

3

For d = 0 we have proven that [BDeM, EJA2011] (a) bi(x) = sinc(n(x − xi)) normalized so that

i bi(x) = 1

(b) lim

n→∞ n αi = 1, αi =

1 bi(x)dx . that is the quadrature process asymptotically converges.

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 40 / 47

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SLIDE 57

An application

bi e sinc Figure: Comparison of bn−1 and sinc(n(x − xn−1)) for n = 10 err = 0.0101

nw =(0.4899,1.1007, 0.9388,1.0475,0.9582,1.0402,0.9582,1.0475,0.9388,1.1007,0.4899) Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 41 / 47

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SLIDE 58

An application

Numerical quadrature The table below shows the quadrature relative errors for d = 0 (left) and d = 3 (right) at different n, for the Runge function. errS=quadrature relative error by using cubic splines n err (d=0) err (d=3) errS 10 3.5e-3 1.1e-2 7.2e-3 30 1.1e-4 1.6e-6 5.9e-5 50 7.6e-6 2.6e-8 3.2e-7 100 3.6e-7 7.9e-10 2.4e-8 150 4.9e-7 1.0e-10 1.5e-9 200 5.4e-7 2.4e-11 6.4e-11

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 42 / 47

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SLIDE 59

An application

Numerical quadrature: an open problem About the quadrature weights: Klein and Berrut have proven numerically that the weights are all positive at least for d ≤ n ≤ 1250 and 0 ≤ d ≤ 5. For other values

  • f d and n, there might be a few negative weights, the number of which increases

slowly with d and n.

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 43 / 47

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SLIDE 60

An application

A Matlab package

  • C. Bandiziol in her degree thesis (University of Padova, Feb. 2015) have
  • rganized all these results and applications in a Matlab package (to be available

soon) that allows Compute FHRI Compute the Lebesgue constants Compute integrals by the Direct and Indirect Rational Quadrature method (BK, BIT 2012) Compute the Least-Square approximation by the FHRI

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 44 / 47

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SLIDE 61

References and announcement

References

1 J.-P. Berrut and H. D. Mittelmann, Lebesgue Constant Minimizing Linear Rational Interpolation of Continuous Functions over the Interval, Computers Math. Appl. 33(6) (1997), 77–86. 2 J.-P. Berrut. Rational functions for guaranteed and experimentally well-conditioned global interpolation. Comput.

  • Math. Appl., 15(1) (1988), 1–16.

3 Michael S. Floater and Kai Hormann, Barycentric rational interpolation with no poles and high rates of approximation,

  • Numer. Math. 107(2) (2007), 315–331.

4

  • L. Bos, S. De Marchi and K. Hormann, On the Lebesgue constant of Berrut’s rational interpolant at equidistant nodes ,
  • J. Comput. Appl. Math. 236 (2011), pp. 504–510.

5

  • L. Bos, S. De Marchi, K. Hormann and G. Klein, On the Lebesgue constant of barycentric rational interpolation at

equidistant nodes, Numer. Math. 121(3) (2012), 461-471. 6

  • K. Hormann, G. Klein,S. De Marchi, Barycentric rational interpolation at quasi-equidistant nodes Dolomites Research

Notes Approx, Vol 5 (2012), 1–6. 7

  • L. Bos, S. De Marchi, K. Hormann and J. Sidon, Bounding the Lebesgue constant of Berrut’s rational interpolant at

general nodes J. Approx. Theory Vol. 169 (2013), 7–22. 8 Cinzia Bandiziol, Interpolante di Floater-Hormann e sue applicazioni, degree thesis - University of Padova (2015), with software in Matlab. Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 45 / 47

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SLIDE 62

References and announcement

4th Dolomites Workshop on Constructive Approximation and Applications Alba di Canazei, 8-13 September 2016 http://events.math.unipd.it/dwcaa16/

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SLIDE 63

References and announcement

THANK YOU FOR YOUR KIND ATTENTION

Stefano De Marchi (DM-UNIPD) Lebesgue constants of rat. interp. Fribourg, November 3, 2015 47 / 47